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The Astrophysical Journal Letters, 836:L1 (6pp), 2017 February 10
https://doi.org/10.3847/2041-8213/836/1/L1
© 2017. The American Astronomical Society. All rights reserved.
On the Accretion Rates and Radiative Efficiencies of the Highest-redshift Quasars
1
Benny Trakhtenbrot1,4, Marta Volonteri2, and Priyamvada Natarajan3
Institute for Astronomy, Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland; [email protected]
2
Institut d’Astrophysique de Paris, UPMC et CNRS, UMR 7095, 98 bis bd Arago, F-75014 Paris, France
3
Department of Astronomy, Yale University, 260 Whitney Avenue, New Haven, CT 06511, USA
Received 2016 November 2; revised 2017 January 6; accepted 2017 January 18; published 2017 February 3
Abstract
We estimate the accretion rates onto the supermassive black holes that power 20 of the highest-redshift quasars, at
z  5.8, including the quasar with the highest redshift known to date—ULAS J1120 at z=7.09. The analysis is
based on the observed (rest-frame) optical luminosities and reliable “virial” estimates of the BH masses of the
quasars, and utilizes scaling relations derived from thin accretion disk theory. The mass accretion rates through the
postulated disks cover a wide range, M˙ disk  4–190 M yr-1, with most of the objects (80%) having
M˙ disk  10–65 M yr-1, confirming the Eddington-limited nature of the accretion flows. By combining our
estimates of Ṁdisk with conservative, lower limits on the bolometric luminosities of the quasars, we investigate
which alternative values of h best account for all the available data. We find that the vast majority of quasars
(∼85%) can be explained with radiative efficiencies in the range h  0.03–0.3, with a median value close to the
commonly assumed h =0.1. Within this range, we obtain conservative estimates of h  0.14 for ULAS J1120 and
SDSS J0100 (at z=6.3), and of 0.19 for SDSS J1148 (at z = 6.41; assuming their BH masses are accurate). The
implied accretion timescales are generally in the range tacc º MBH M˙ BH  0.1–1 Gyr , suggesting that most
quasars could have had ~1–10 mass e-foldings since BH seed formation. Our analysis therefore demonstrates that
the available luminosities and masses for the highest-redshift quasars can be explained self-consistently within the
thin, radiatively efficient accretion disk paradigm. Episodes of radiatively inefficient, “super-critical” accretion may
have occurred at significantly earlier epochs (i.e., z  10 ).
Key words: black hole physics – galaxies: active – galaxies: nuclei – quasars: general
for advection-dominated flows, characteristic of significantly
sub-Eddington accretion (e.g., Narayan & Yi 1995), or “supercritical” accretion flows (e.g., Paczyńsky & Wiita 1980).
Additionally, in these regimes, the luminosity is not proportional to the accretion rate, but to the accretion rate squared and
the logarithm of the accretion rate, respectively.
Importantly, the relevance of either low-h mechanisms or the
assumption of a universal h =0.1 to the observed population
of the earliest known quasars, are not yet established.
In this Letter, we use insights from thin accretion disk theory
and basic observables of some of the highest-redshift quasars
known to date, at z  6, to investigate the mass accretion rates
and the corresponding radiative efficiencies powering these
systems. This work assumes a cosmological model with
WL = 0.7, WM = 0.3, and H0 = 70 km s-1 Mpc-1.
1. Introduction
The existence of luminous quasars as early as z ~ 6–7
suggests that supermassive black holes (SMBHs) with masses
of order MBH ~ 109 M were in place less than 1 Gyr after the
Big Bang. This is explicitly shown by observations that trace
the gas dynamics in the close vicinity of the accreting SMBHs
(Kurk et al. 2007; Willott et al. 2010; De Rosa et al. 2011,
2014; Trakhtenbrot et al. 2011). Such masses require
continuous, exponential growth at the Eddington limit,
L LEdd = 1, for almost the whole age of the universe at that
time, and these conditions may not be necessarily ubiquitous
among early SMBHs (e.g., Treister et al. 2013; Habouzit et al.
2016; Trakhtenbrot et al. 2016; Volonteri & Reines 2016).
The ability of SMBHs to grow to MBH ~ 109 M depends,
critically, on the radiative efficiency of the accretion flow—
defined as h º L bol M˙ acc c 2 , where Ṁacc is the mass inflow rate
and L bol is the emerging bolometric luminosity. Most
calculations of early BH growth assume a universal value of
h  0.1, relying on the ensemble properties of quasars and relic
SMBHs across all cosmic epochs.
In reality, however, the role of h in early BH growth is more
complex. Individual systems should have various h , as indeed
suggested by the observed range of BH spins (see, e.g., the
review by Reynolds 2014 and also Davis & Laor 2011;
Trakhtenbrot 2014; Capellupo et al. 2016). The value of
h  0.1 is within the range expected in optically thick,
geometrically thin accretion disks, where h ~ 0.04–0.32,
depending on the BH spin. However, we recall that η would
be much lower for geometrically thick accretion disks, such as
4
2. Method, Sample, and Data
The goal of the present study is to test whether the currently
available data for the highest-redshift quasars known to date
can be self-consistently explained within the thin accretion disk
model and to estimate the corresponding accretion rates (Ṁdisk )
and radiative efficiencies (h ). The method we use is based on
two fundamental assumptions: that the SMBHs we study are
powered by thin accretion disks and that their masses are
reliably known. In thin-disk accretion flows, the rest-frame
optical continuum emission (l rest  4500 Å), originating
primarily from the outer disk, follows a power-law form,
L n µ (MBH , M˙ disk )2 3 n1 3 (see, e.g., Davis & Laor 2011 and
references therein). Rewriting Ṁdisk in terms of MBH and the
(monochromatic) continuum luminosity along this power-law
Zwicky postdoctoral fellow.
1
The Astrophysical Journal Letters, 836:L1 (6pp), 2017 February 10
Trakhtenbrot, Volonteri, & Natarajan
tail provides
⎛ lLl,45 ⎞3
⎟
M˙ disk  2.4 ⎜
⎝ cos i ⎠
2
⎛ l ⎞2 -1
⎜
⎟ M
M yr-1,
⎝ 5100 Å ⎠ 8
obtained by cross-matching our sample with the AllWISE data
release, within 5″ of the optical coordinates of the sources. In
all cases where both Spitzer and WISE measurements are
available, we preferred the higher spatial resolution and
sensitivity Spitzer data. The first two bands of the IRAC or
WISE cameras have effective wavelengths of roughly 3.6 and
4.5 μm. For our sample’s redshift range, these correspond to
rest-frame wavelengths of about 4850–5340 and 6065–6675 Å,
respectively. We verified that none of our sources is affected by
blending with neighboring WISE sources. All Spitzer and WISE
measurements were converted to flux densities using standard
procedures (i.e., Wright et al. 2010; Jarrett et al. 2011).
Monochromatic luminosities were calculated assuming the
Mg II -based redshifts reported in the aforementioned NIR
studies (see Table 1).
Our final sample includes 20 quasars with reliable estimates
of MBH , 9 with Spitzer data and the remaining 11 with WISE
data. We verified that none of the choices we made in
compiling the data set has significant effects on our results. The
heterogeneous nature of our sample—drawn from several
surveys of varying depth, and our obvious focus on vigorously
accreting SMBHs at this extremely high redshift regime, mean
that our sample is most probably not representative of the entire
population of (active) SMBHs at z  6.
We calculated Ṁdisk for the 20 quasars through Equation (1),
using the (re-calculated) Mg II -based MBH estimates, and the
monochromatic luminosities observed in either of the IR bands
(i.e., ∼3.6 and 4.5 μm, hereafter L 3.6 and L 4.5, respectively).
We list all the quantities relevant for the present analysis,
including the chosen sources of all measurements, in Table 1.
(1 )
where λ is the (rest-frame) wavelength at which the continuum
is measured, lLl,45 º lLl 10 45 erg s-1 denotes the monochromatic luminosity, and M8 º MBH 108 M. cos i represents the
inclination between the line of sight and the polar axis of the
disk (here we adopt cos i = 0.8, as appropriate for broad-line
quasars). The derivation of this expression is discussed in detail
in, e.g., Davis & Laor (2011) and Netzer & Trakhtenbrot
(2014). This approach was used in several recent studies of
accretion flows for samples of quasars to z  3.5 (e.g., Bian &
Zhao 2003; Davis & Laor 2011; Wu et al. 2013;
Trakhtenbrot 2014).
At z  6 the estimation of Ṁdisk through Equation (1)
necessitates flux measurements at ~3.5–10 μm and K-band
spectroscopy of the Mg II λ2798 broad emission line (for MBH
estimation; see, e.g., Trakhtenbrot & Netzer 2012,
hereafter TN12, and references therein). Here, we focus only
on those z  6 quasars for which such data are publicly
available.
Compiling all the z  6 quasars for which Mg II -based
estimates of MBH are available from NIR spectroscopy, we find
35 objects in the studies of Iwamuro et al. (2004), Jiang et al.
(2007), Kurk et al. (2007, 2009), Willott et al. (2010),
Venemans et al. (2013), De Rosa et al. (2014), and Venemans
et al. (2015). We additionally include the well-studied z  6.4
quasar SDSS J1148 (from Barth et al. 2003) and the highestredshift quasar known to date, ULAS J1120 (z = 7.085;
Mortlock et al. 2011). We finally include the extremely
luminous and massive quasars J0100+2802 (z = 6.3; Wu
et al. 2015) and J0306+1853 (z = 5.363; Wang et al. 2015).
Although J0306 is at a lower redshift, we include it to test if the
MIR-based high-redshift quasar selection methods may be
unveiling a distinct population of SMBHs. Throughout this
work, we highlight the results obtained for these four quasars of
interest, but note here that they should be viewed as part of an
ensemble of quasars.
For some of the quasars there are multiple published NIR
spectra and/or Mg II profile measurements. Whenever possible, we have consistently used the detailed measurements
performed by De Rosa et al. (2011). These replace the
measurements provided by Iwamuro et al. (2004), Jiang et al.
(2007), Kurk et al. (2007; except for J0836), and Kurk et al.
(2009). For sources with multiple sets of measurements in De
Rosa et al. (2011), we selected those with the smaller
uncertainties on FWHM (Mg II). Using the measurements of
L 3000 and FWHM (Mg II) reported in the selected studies (and
adjusted for our chosen cosmological model), we re-calculated
all MBH estimates following the prescription of TN12 (see also
Shen et al. 2011). These accurate estimates of MBH are known
to carry systematic uncertainties of up to 0.5 dex (TN12 and
references therein).
We then compiled rest-frame optical photometric data,
obtained with the Spitzer and WISE IR space telescopes. For
nine of the quasars, we use Spitzer/IRAC data reported in the
studies of Jiang et al. (2006), Leipski et al. (2014), and Barnett
et al. (2015; for ULAS J1120). Whenever possible, we used the
Leipski et al. (2014) measurements for homogeneity. For 11
quasars we use WISE measurements in the W1 and W2 bands,
3. Results
Figure 1 presents the derived accretion rates through the
disks, Ṁdisk , for all the sources in our sample, and based on the
two different (rest-frame) optical luminosities. The accretion
rates we obtain using L 4.5 are in the range M˙ disk 
3.6–187 M yr-1, with 16 of the quasars (80%) having
M˙ disk  10–65 M yr-1. We note that some of the variance
in ṀBH in our sample may be attributed to the significant
systematic uncertainties in MBH , and the form of Equation (1).
Notwithstanding this limitation, we obtain M˙ disk =
11.4 M yr-1 for ULAS J1120, while for SDSS J1148 and
SDSS J0100 we find M˙ disk  16.3 and 54.6 M yr-1, respectively. The accretion rate we find for SDSS J0306 is in
excellent agreement with that of SDSS J0100, which is
expected given the very similar masses and continuum
luminosities of the two quasars (both within 0.1 dex). The
L 3.6 -based Ṁdisk estimates are systematically lower than those
based on L 4.5, by about 0.19 dex (median value). This is
probably due to the fact that the 4.5 μm band includes the
strong broad Hα line emission, which is expected to be
stronger by a factor of ∼3 compared with the Hβ line, covered
in the 3.6 μm band data of most sources. Moreover, the 3.6 μm
band is probing the continuum emission in a spectral regime
where the power-law approximation may no longer be valid
(particularly at high MBH ; see, e.g., Davis & Laor 2011; Netzer
& Trakhtenbrot 2014). In what follows, we focus on the
L 4.5-based estimates of Ṁdisk , as these would result in more
conservative constraints on h (i.e., lower limits; see below).
We next use the estimates of Ṁdisk and the observed
luminosities of our quasars to investigate the range of—or
lower limits on—h , which would be consistent with all the data
2
The Astrophysical Journal Letters, 836:L1 (6pp), 2017 February 10
Trakhtenbrot, Volonteri, & Natarajan
Table 1
Observed and Derived Properties
za
log L 3000 b
(erg s-1)
FWHM (Mg II)
(km s-1)
NIR
Ref.c
log MBH d
(M)
log L 3.6 e
(erg s-1)
log L 4.5e
(erg s-1)
MIR
Ref.f
Ṁdisk g
(M yr-1)
hh
J1120+0641
J1148+5251
J0100+2802
7.097
6.407
6.3
46.48
46.79
47.50
4411
5352
5130
2
1
4
9.58
9.93
10.34
46.50
46.73
47.31
46.36
46.65
47.24
3
1
2
11.4
16.3
54.6
0.144
0.189
0.139
J0306+1853
5.363
47.31
5722
5
10.32
47.20
47.15
2
55.4
0.126
J0050+3445
J0836+0054
J0353+0104
J0842+1218
J2348−3054
J0305−3150
P036+03
P338+29
J0005−0006
J1411+1217
J1306+0356
J1630+4012
J0303−0019
J1623+3112
J1048+4637
J1030+0524
6.253
5.81
6.072
6.069
6.889
6.605
6.527
6.658
5.844
5.903
6.017
6.058
6.079
6.211
6.198
6.302
46.55
46.93
46.42
46.47
45.99
46.24
46.68
46.11
46.00
46.57
46.34
46.25
46.01
46.36
47.28
46.36
4360
3600
3682
3931
5446
3189
3500
6800
1036
2824
3158
3366
2307
3587
3366
3449
6
7
1
1
2
2
3
3
1
1
1
1
1
1
1
1
9.61
9.67
9.38
9.47
9.46
9.15
9.50
9.72
8.02
9.24
9.20
9.20
8.72
9.32
9.83
9.28
46.42
46.91
46.32
46.45
46.11
46.07
46.44
46.02
46.02
46.45
46.39
46.11
46.00
46.43
46.60
46.45
46.37
47.01
46.29
46.43
46.10
46.05
46.28
46.07
46.03
46.55
46.34
46.07
46.02
46.45
46.54
46.43
2
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
14.6
124.4
18.4
24.6
6.7
12.6
12.7
3.6
186.6
64.8
33.9
13.3
33.3
35.2
15.2
35.2
0.131
0.032
0.079
0.065
0.083
0.077
0.197
0.199
0.003
0.031
0.036
0.075
0.017
0.036
0.444
0.036
Object
Notes.
a
Redshift measured from the best-fit model of the Mg II line.
b
Monochromatic luminosity (lLl ) at rest-frame wavelength of 3000 Å, obtained from the best-fit model of the Mg II emission line complex.
c
References for NIR spectral analysis and Mg II measurements: (1) De Rosa et al. (2011); (2) De Rosa et al. (2014); (3) Venemans et al. (2015); (4) Wu et al. (2015);
(5) Wang et al. (2015); (6) Willott et al. (2010); (7) Kurk et al. (2007).
d
BH mass, estimated using the Mg II line and the TN12 prescription.
e
Monochromatic luminosities (lLl ) at observed-frame wavelengths of ∼3.6 and 4.5 μm.
f
References for MIR photometry: (1) Spitzer (Leipski et al. 2014; including detections by Jiang et al. 2006); (2) WISE cross-match (see also Wu et al. 2015 for J0100
and Wang et al. 2015 for J0306); (3) Spitzer (Barnett et al. 2015).
g
Obtained using Equation (1), L 4.5 and MBH .
h
Obtained using the bolometric corrections of TN12, and the L 4.5-based Ṁdisk .
To further test the range of h consistent with the data, we
also calculated L bol using the L 3000 -dependent bolometric
corrections of TN12, which for the sample in hand are in the
range fbol (3000 Å) ~ 2–3.2. Coupling these L bol estimates
with the L 4.5-based Ṁdisk estimates, we obtain h estimates in the
range of h  0.003–0.44 (see Table 1 and the dashed line in
Figure 2). In this case, 12 of the 20 quasars (60%) have h
within the range of values expected for thin disks. Only two
objects have h < 0.03, and only one has h > 0.3. Focusing
again on the three quasars mentioned above, we obtain
h  0.14, 0.19, and 0.14, for ULAS J1120, SDSS J1148, and
SDSS J0100, respectively. For SDSS J0306 at z=5.36 we find
h  0.13, again highly consistent with SDSS J0100.
We note that these latter L 3000 -based estimates of L bol and h
may also be considered conservative, as the TN12 bolometric
corrections we use are, again, significantly lower than those
commonly used for samples of high-z quasars (by factors of
∼2; see, e.g., the compilation of Runnoe et al. 2012). If we had
instead used the fbol (3000 Å) suggested by Runnoe et al.
(2012), the resulting h estimates would have been higher
by ∼30%.
Figure 3 presents our two sets of (conservative) h estimates
of h against MBH . The apparent trend of increasing h with
increasing MBH is mainly driven by the explicit dependence of
Equation (1) on MBH , and then on the fact that h µ 1 M˙ disk (at
fixed luminosity). It is therefore expected that rest-frame UV-
available for the z  6 quasars, following h = L bol M˙ disk c 2 .
Given the data in hand, L bol can only be estimated by utilizing
bolometric corrections and monochromatic luminosities (unlike
the analysis of Davis & Laor 2011).
We focus on conservative, lower limits on h , which can be
obtained assuming L bol = 3 ´ L 3.6 (3 ´ lLl [4930 Å]). This
choice is very similar to the one made in Trakhtenbrot (2014;
L bol = 3 ´ lLl [5100 Å]). We consider it to provide a lower
limit on the real L bol since it reflects a bolometric correction
that is much smaller, by at least a factor of ∼2, than those used
in many other studies of MBH and L LEdd in high-redshift
quasars (see, e.g., Runnoe et al. 2012). Since we seek to derive
lower limits on h , we further use the higher, L 4.5-based
estimates of Ṁdisk . As h µ 1 M˙ disk µ MBH , our h estimates
inherit the systematic uncertainties on MBH (see above).
Figure 2 shows the cumulative distribution of the conservative
constraints on h we obtain, which are in the range
h  0.003–0.2. Most quasars have lower limits on h that are
consistent with what is expected for thin disks, and only three
quasars have lower limits on h that are below 0.03. For the
three quasars of particular interest we find conservative lower
limits of h  0.14, 0.17, and 0.2 (for ULAS J1120,
SDSS J1148, and SDSS J0100, respectively). For SDSS J0306
we find h  0.15, consistent with the z  6 quasars and
particularly with SDSS J0100, which has a very similar
BH mass.
3
The Astrophysical Journal Letters, 836:L1 (6pp), 2017 February 10
Trakhtenbrot, Volonteri, & Natarajan
Figure 3. Estimates of radiative efficiencies h , vs. BH mass, MBH , for the 20
quasars compiled in our sample. For each quasar, we show two different
estimates of h , with the lower value being our most conservative estimate
(based on L bol = 3 ´ L 3.6 and the L 4.5-based Ṁdisk estimates), and the higher
value derived through the L 3000 -based L bol estimates (the TN12 bolometric
corrections). The real h may be higher than what these two sets of estimates
suggest. The most massive BHs in our sample show high radiative efficiencies,
h  0.2 . The apparent trend of increasing h with increasing MBH is likely
driven by the form of Equation (1). We cannot rule out that SMBHs with
MBH  1010 M and h  0.1 exist, but are not (yet) observed.
Figure 1. Cumulative distribution function of our estimates of accretion rates
through the disks, Ṁdisk , based on Equation (1). The dashed and solid lines
represent the estimates based on L 3.6 and L 4.5, respectively. The systematically
higher L 4.5-based estimates of Ṁdisk result in more conservative constraints on
h (i.e., lower values) and shorter accretion timescales.
enough number of such extreme, yet-to-be-observed objects
may further constrain models of BH seed formation and early
growth (e.g., Agarwal et al. 2013).
We next turn to estimate the accretion timescales of the
SMBHs powering the quasars, tacc ≡MBH /ṀBH , which may be
considered as the typical BH mass e-folding timescales. Here,
we derive and compare two sets of tacc estimates made available
by the data, and following two approaches for estimating
M˙ BH = (1 - h ) M˙ disk . First, we follow the common procedure
of using L LEdd and a fixed radiative efficiency (i.e., h = 0.1),
to
derive
accretion
timescales
through
tacc  0.4 (h 1 - h ) Gyr . The tacc estimates thus obtained are
shown in the left panel of Figure 4, plotted against the age of
the universe (at the observed epoch). Most objects have
tacc ~ 0.1–1 Gyr , and could have had between ~1–10 mass efoldings of MBH . Second, we use our Ṁdisk estimates, which
provide M˙ BH = (1 - h ) M˙ disk , and our conservative estimates
of h . Due to the dependence of h on Ṁdisk in our analysis, we
note that these estimates can be expressed as
tacc = MBH (M˙ disk - L bol c 2 ). As before, we adopt the
L 3000 -based estimates of L bol , and either the L 3.6 - or L 4.5-based
estimates of Ṁdisk . The accretion timescales we obtain through
this procedure are presented in the right panel of Figure 4. The
shorter, L 3.6 -based timescales, are generally in the range of
tacc ~ 0.01–2 Gyr , with 18 of the quasars (90%) having
tacc  0.03–0.8 Gyr . The extremely high-Ṁdisk quasar J0005
(M˙ disk > 100 M yr-1) has the shortest accretion timescale,
tacc  0.5 Myr (cf. ∼0.02 Gyr obtained from L LEdd ). For the
three z > 6 quasars of interest, ULAS J1120, SDSS J1148, and
SDSS J0100, we find tacc = 0.36, 0.71 and 0.67 Gyr,
Figure 2. Cumulative distribution function of radiative efficiency estimates, h .
The solid line traces the values obtained using the conservative assumption of
L bol = 3 ´ L 3.6 (roughly L bol  3 ´ L 5100 ) and the generally higher
L 4.5-based estimates of Ṁdisk . The dashed line traces the h estimates based
on L bol (L 3000 ), using the TN12 bolometric corrections. The vertical dashed
lines mark range of radiative efficiencies expected for thin accretion disks
around spinning BHs (0.038  h  0.32 ). The dotted vertical line
marks h =0.1.
optical surveys of a given depth would miss high-MBH but lowh objects (see also Bertemes et al. 2016). In this context,
extremely massive objects like SDSS J0100 and SDSS J0306,
which have MBH  1010 M, may represent the high-h end of a
much larger population of high-mass BHs at z ~ 5–6. A large
4
The Astrophysical Journal Letters, 836:L1 (6pp), 2017 February 10
Trakhtenbrot, Volonteri, & Natarajan
Figure 4. Different estimates of BH accretion timescales, tacc = MBH M˙ BH , vs. cosmic epoch. Left: “standard” tacc timescale estimates, obtained from L LEdd and
assuming h = 0.1. Right: tacc estimates obtained from Ṁdisk and the L 3000 -based estimates of L bol (i.e., equivalent to using our h estimates; see Table 1). For each
source, we plot the timescales obtained from both the L 3.6 - and L 4.5-based estimates of Ṁdisk .
respectively. The ultramassive z  5.3 quasar SDSS J0306 has
tacc = 0.66 Gyr . The Ṁdisk -based estimates of tacc suggest that
some quasars could have had as many as 50 BH mass efoldings.
We stress, again, that the trends of decreasing tacc with epoch
seen in Figure 4 are due to the way the original samples of
quasars were selected and identified (i.e., their brightness and
luminosity), and the form of the prescriptions we use here.
The two sets of tacc estimates are generally in good
agreement, with differences being within a factor of ∼2 for
14 (70%) of our quasars, and even within a factor of ∼1.5 for 8
(40%) of the quasars. The lack of a significant systematic offset
between the two timescale estimates reflects the fact that our h
estimates bracket the “standard” value of h = 0.1. We conclude
that for our sample of z  6 quasars, the simpler L LEdd -based
growth timescale estimates are broadly consistent with those
derived through our more elaborate approach.
limits) on the radiative efficiencies that are in the range
0.04  h  0.32—that is, within the range expected for
accretion through a thin disk onto rotating BHs. Our more
conservative estimates suggest h  0.05 for most objects.
Thus, it appears that all the data available for quasars at z  5.8
are consistent with such radiatively efficient accretion flows.
Moreover, since our analysis provides lower limits on h , it is
possible that the real h of the quasars under study would differ
substantially with the expectations of radiatively inefficient
accretion flow models.
We stress that this result is not a trivial consequence of the
observables and methodology we adopt here (i.e., Equation (1)).
For example, the study of Trakhtenbrot (2014)—applying the
same methods as the ones used here to a sample of high-MBH
quasars at 1.5  z  3.5—found extremely high values of h ,
which in many cases exceeded h  0.4. Among the conservative h estimates, we find h  0.15 for ULAS J1120—the
highest-redshift quasar known to date (z=7.1), and h  0.2
for SDSS J1148 (at z=6.4). On the other hand, many other
z  6 quasars have 0.05  h  0.1—below the standard,
universal radiative efficiency assumed in many studies of the
AGN population. The limited size of our sample prevents us
from determining whether these h estimates represent the
scatter within the quasar population, or only trace a few
extreme cases.
The accretion timescales of the SMBHs under study, derived
assuming the Ṁdisk and h estimates, are consistent with those
derived from L LEdd and the universal h =0.1 assumption,
allowing for ~1–100 mass e-foldings. This further justifies the
usage of the simpler tacc estimates in cases where only L LEdd
is available (i.e., when rest-frame optical luminosities are
unavailable). However, this assumption would naturally
neglect the fact that any population of accreting SMBHs is
expected to have a range of h .
4. Discussion and Conclusion
The main point of our analysis is to investigate whether the
data available for some of the highest-redshift quasars known
to date can be accounted for, self-consistently, within the
generic model of a radiatively efficient, thin accretion disk.
We found that the accretion rates through the postulated thin
disks are generally in the range of M˙ disk ~ 1–100 M yr-1.
These accretion rates are consistent with the systems being
Eddington-limited, at the observed epoch. However, if one
assumes that these accretion rates were sustained at earlier
epochs, when the BH masses were considerably lower, this
would imply super-Eddington accretion rates, which may be
sustained under certain gas configurations, and lead to a fast
buildup of BH mass (e.g., Volonteri et al. 2015).
We showed that the available data for most of the z  6
quasars can be explained with conservative estimates (lower
5
The Astrophysical Journal Letters, 836:L1 (6pp), 2017 February 10
Trakhtenbrot, Volonteri, & Natarajan
Within the standard thin-disk framework, radiative efficiencies are closely linked to BH spin, a* (in normalized units). The
range of h we find corresponds to the entire possible range of
-1  a*  1, and the typical (median) h of our sample
corresponds to a*  0.7—again consistent with what is
expected from the assumption of a universal h =0.1. As the
quasars of interest in our sample (ULAS J1120, SDSS J1148,
and SDSS J0100) have h > 0.1, they correspond to rather high
spins, a*  0.9. These, in turn, are consistent with what is
found for low-redshift, low-luminosity AGNs (Reynolds 2014
and references therein) and for higher-luminosity, higher-MBH
quasars at higher redshifts (Reis et al. 2014; Trakhtenbrot 2014;
Capellupo et al. 2016). This supports a scenario in which the
z  6 SMBHs grew through coherent accretion flows (e.g.,
Dotti et al. 2013; Volonteri et al. 2013 and references therein).
This “spin up” scenario appears highly plausible, given the
high duty cycle of accretion required for the fast BH growth
at z > 6 .
Several recent studies highlighted the possibility that z  6
quasars could have grown through “super-critical” accretion to
reach their high BH masses (Volonteri & Rees 2005;
Alexander & Natarajan 2014; Volonteri et al. 2015). The more
extreme cases of such accretion, in slim disks, may result in
m˙ m˙ Edd ~ 100 and h ~ 0.01, essentially regardless of the BH
spin (see, e.g., Sadowski 2009; Madau et al. 2014; McKinney
et al. 2014; Volonteri et al. 2015, but see McKinney
et al. 2015). Our analysis confirms that such super-critical
growth episodes should have occurred, if at all, in the yet
earlier universe, when BHs were smaller, to alleviate the
requirement of continuous growth, which may not be realistic.
The data currently available for the highest redshift quasars,
namely, in the (rest-frame) UV and optical, as well as the data
that may become available in the foreseeable future, cannot
directly distinguish between radiatively efficient and inefficient
accretion. Models of such flows require further study, with an
emphasis on the observables that are relevant for faint, z > 6
sources. For instance, super-critical episodes are likely
accompanied by the production of powerful jets (McKinney
et al. 2014; Sadowski et al. 2014; Sadowski & Narayan 2016).
At high-z such jets should be detectable in X-rays rather than as
extended radio sources (Ghisellini et al. 2015). This, together
with the limitations present in the (rest-frame) UV regime due
to IGM absorption, highlight the importance of the X-ray
regime, where surveys of ever-increasing depth (e.g., in the
CDF-S field) may provide key insights into the assembly of the
earliest SMBHs.
the National Science Foundation. This work made use of the
MATLAB package for astronomy and astrophysics
(Ofek 2014).
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We thank K. Schawinski for beneficial discussions. This
work was performed in part at the Aspen Center for Physics,
which is supported by National Science Foundation grant
PHY-1066293. M.V. acknowledges funding from the European Research Council under the European Community’s
Seventh Framework Programme (FP7/2007–2013 grant agreement No. 614199, project “BLACK”). P.N. acknowledges
support from TCAN grant with award number 1332858 from
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